Use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field F = 4zi +5xj + 2yk across the surface S: r(r,0) = r cos 0i+r sin 0j + (16-r²) k, 0≤r≤4, 0≤0≤2 in the direction away fromthe origin.The flux of the curl of the field F is(Type an exact answer, using as needed.)

Answered: Image /qna-images/answer/b04aa08f-a5a9-4273-b492-6b12296a5c11.jpg

Answered: Use the surface integral in Stokes'…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field F across the surface S in the direction away from the origin. F = 3yi + (5 - 5x)j + (z² − 2)k S: r(0,0) = (√5 sin & cos 0) i + (√√5 sin þ sin 0)j + (√5 cos $) k, 0≤þ≤ñ/2, 0≤0≤2 The flux of the curl of the field F across the surface S in the direction of the outward unit normal n is (Type an exact answer, using as needed.)
B2: Assume K is a body (Area) that is coherent. Let C be the curve that is parametised by the following equation (tan () .cos"(), sin°(1) ) osisa. Assume the following vector field thet is well defined in "K" and "a" is the value that gives Fa potential V(x, y, z). axz 1 + 2r, x²z+ y + 2y, + 2z 1²z + y° F(x, y, z) = 1²z + y_ a) Calculate a) b) Calculate the lineintegral F - dr.
Let vector field F be given, C the curve with counterclockwise orientation looking from the direction of the positive z-axis. Calculate the line integral of the dot prpoduct F and dR.
Use the surface integral in Stokes' Theorem to calculate the circulation of the field F around the curve C in the indicated direction. F=yi+xzj+x²k C: The boundary of the triangle cut from the plane 5x+y+z=5 by the first octant, counterclockwise when viewed from above. The circulation is (Type an integer or a fraction.)
Prove Stokes theorem for the vector field P over the triangle area bounded by this path by the path indicated by the arrows in the xy planet in the figure.
Use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field F across the surface S in the direction away from the origin. F = 4yi + (5-2x)j + (z² - 2)k S: r(0, 0) =(√6 sin cos 0) i + ( √5 sin ô sin 0)j + (√√6 cos ¢) k, 0≤ ≤r/2, 0≤0≤2r The flux of the curl of the field F across the surface S in the direction of the outward unit normal n is (Type an exact answer, using as needed.)
Compute the line integral of the vector field ?=⟨7?,−7?⟩ over the circle ?^2+?^2=1 oriented clockwise
Compute the flux of the vector field F = 2xi + 3yj through the cylindrical surface shown below, if a = 4 and b = 4 and it is oriented away from the z- axis. flux =

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